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AFFINE HECKE ALGEBRAS and THEIR GRADED VERSION 0.1. Let JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 2, Number 3, July 1989 AFFINE HECKE ALGEBRAS AND THEIR GRADED VERSION GEORGE LUSZTIG Dedicated to Sir Michael Atiyah on his sixtieth birthday INTRODUCTION 0.1. Let Hvo be an affine Hecke algebra with parameter Vo E C* assumed to be of infinite order. (The basis elements ~ E Hvo corresponding to simple reflections s satisfy (~+ 1)(Ts - v~C(s») = 0, where c(s) EN depend on s and are subject only to c(s) = c(s') whenever s, Sf are conjugate in the affine Weyl group.) Such Hecke algebras appear naturally in the representation theory of semisimple p-adic groups, and understanding their representation theory is a question of considerable interest. Consider the "special case" where c(s) is independent of s and the coroots generate a direct summand. In this "special case," the question above has been studied in [I] and a classification of the simple modules was obtained. The approach of [1] was based on equivariant K-theory. This approach can be attempted in the general case (some indications are given in [5, 0.3]), but there appear to be some serious difficulties in carrying it out. 0.2. On the other hand, in [5] we introduced some algebras H,o' depending on a parameter roE C , which are graded analogues of Hvo . The graded algebras H '0 are in many respects simpler than Hvo' and in [5] the representation theory of H,o is studied using equivariant homology. Moreover, we can make the machinery of intersection cohomology work for us in the study of H '0 ' while in the K-theory context of Hvo it is not clear how to do this. In particular, the difficulties mentioned above disappear when Hvo is replaced by H '0 • For this reason, it seemed desirable to try to connect the representation the- ories of Hvo and H,o' In this paper, we shall prove that the classification of simple Hvo -modules can be reduced to the same problem, where Hvo is replaced essentially by H,o' This makes it possible to study the representation theory of Hvo without K-theory but with the aid of H,o' (This is analogous to studying the representations of a Lie group using the theory of Lie algebras.) It should Received by the editors March 14, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A89, 16A03. © 1989 American Mathematical Society 0894-0347/89 $ 1.00 + $.25 per page 599 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 600 GEORGE LUSZTIG be pointed out that this approach fails when Vo is a root of 1, other than 1. This is a very interesting case which has been studied very little. 0.3. In an unpublished work, Bernstein (partly in collaboration with Zelevin- skii) described a large commutative subalgebra of Hvo and the center of Hvo (in the "special case" above). We extend these results to the general case. A number of new difficulties arise, most of them caused by the simple coroots which are divisible by 2. This extension is done in §3, based on the preparation of §§1 and 2. In §4, we introduce a filtration of an affine Hecke algebra and show that the corresponding graded algebra is the one introduced in [5]. §§5-7 are again preparatory. In §8, we show that the completion of an affine Hecke algebra with respect to a maximal ideal of the center is isomorphic to the ring of n x n matrices with entries in the completion of a (usually smaller) affine Hecke alge- bra. In §9, we define a natural homomorphism from an affine Hecke algebra to a suitable completion of its graded version, which becomes an isomorphism when the first algebra is completed. This homomorphism is of the same nature as the Chern character from K-theory to homology. In §1O, we combine the results of §§8 and 9 to get our main result (see 10.9), comparing the representation theory of an affine Hecke algebra with that of a graded one. 0.4. Most of the work on this paper was done during the Fall of 1988 when I enjoyed the hospitality and support of the Institute for Advanced Study, Prince- ton. I acknowledge partial support of N.S.F. grant DMS-8610730 (at lAS) and DMS-8702842. CONTENTS 1. Root systems and affine Weyl groups 2. The braid group 3. The Hecke algebra 4. The graded Hecke algebra 5. The elements tw and Tw 6. Some braid group relations 7. Completions 8. First reduction theorem 9. Second reduction theorem 10. On simple H -modules References 1. ROOT SYSTEMS AND AFFINE WEYL GROUPS 1.1. A root system (X,Y,R,R,n) consists of (a) two free abelian groups of finite rank X, Y with a given perfect pairing (,}:XxY--+Z, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use AFFINE HEeKE ALGEBRAS AND THEIR GRADED VERSION 601 (b) two finite subsets ReX, R c Y with a given bijection R +-+ R denoted 0: +-+ a , and (c) a subset II cR. These data are subject to requirements (d)-(f) below. (d) (0: ,a} =2 for all o:ER. (e) For any 0: E R, the reflection sa: X -t X, X t-+ X - (x, a}o: (resp. Sa: Y -t Y, y -t y - (o:,y}a) leaves stable R (resp. R). (f) Any 0: E R can be written uniquely as 0: = E pEn na,p • p, where na ,p E Z are all ~ 0 or all :::; O. (Accordingly, we say that 0: E R+ or 0: E R- .) Throughout this paper we will fix a root system (X, Y ,R, R, ll) . The subgroup of GL(X) generated by all sa (0: E R) may be identified with the subgroup of GL(Y) generated by all sa (0: E R) by w -t tw-I . This is the Weyl group Wa of the root system. It is a finite Coxeter group on generators {salo: E ll}. 1.2. Consider the partial order:::; on R defined by a l :::; a 2 <:} a 2 - a l is a linear combination with integer ~ 0 coefficients of elements of {alo: E ll} . Let llrn be the set of all PER such that /J is a minimal element for :::;. 1.3. We shall assume throughout the paper that (X, Y ,R, R, ll) is reduced, i.e., 0: E R => 20: (j. R . 1.4. Let W be the semidirect product Wa' X. Its elements are wax (w E f ,-1, W, X E X) with multiplication given by wax. w' aX = ww' aW (X)+X; a is a fixed symbol. W acts on Y x Z by wax: (y, k) -t (w(y) , k - (x, y}). Let ~ ~+ ~ R = R U R- c Y x Z be defined by R+ = {(a,k)lo: E R, k > O} U {(a , 0)10: E R+}, R- = {(a,k)lo:ER, k <O}U{(a,O)IO:ER-}. Then R is a W -stable subset of Y x Z . Define I: W -t N by (a) I(wax) = #{A E R+I(wax)(A) E R-} -, = L l(x,a}+ll+ L l(x,a}l· Let fi = {(a ,0)10: E ll} U {(a, 1)10: E llrn} c R+, S = {salo: E ll} U {saaalo: E llrn} C W. We have an obvious bijection fi +-+ S (A +-+ SA)' An element of S maps the corresponding element of fi to its negative and it maps the complement of this License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 602 GEORGE LUSZTIG element in R+ into itself. It follows that for z E W, A E fi, we have (b) I(SAZ) = { I(z) + 1, if z-I(A) E R+, I(z) - 1, if z-I(A) E R- . It is clear that (c) I(w) = I(w-I) (w E W). We deduce that for x EX, a E TI, we have (dl) {x ,a} > 0 => l(soaX) = I(ax) + 1, l(aXso) = I(ax) - 1. (d2) {x ,a} < 0 => l(soaX) = I(ax) - 1, I(axso) = I(ax) + 1. (d3) {x ,a} = 0 => I(soax) = I(x) + 1, l(aXso) = I(x) + 1. (e) Let Xdom = {x E XI{x ,a} ~ 0, Va E TI} = {x E XI/(soax) = I(ax) + 1, Va E TI}. We have, using (a) and (e), (f) WE Wa, x E Xdom => I(ax) = EoER+{X ,a}, I(wax) = I(w) + I(ax). It follows that I x X' X X' (g) X,X EXdom=>/(a ·a )=/(a )+/(a ). 1.5. Let Q be the subgroup of X generated by R. The subgroup WoQ of W is a Coxeter group with S as the set of simple reflections, the length function being the restriction of I. This subgroup is normal in Wand admits a complement n = {w E Wlw(ll) = fi} = {w E WI/(w) = O}. It is known that n is an abelian group, isomorphic to X / Q . 1.6. The notion of direct sum of root systems is defined in an obvious way. We say that (X, Y, R, R, TI) is primitive if either (a) or (b) below holds. (a) a ¢ 2Y for all a E R. (b) There is a unique a E TI such that a E 2Y; moreover, {w(a)lw E Wo} generate X. Lemma 1.7. In general, (X, Y,R,R,TI) is (uniquely) a direct sum a/primitive root systems.
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